Lectures on Nuclear and Hadron Physics · Lectures on Nuclear and Hadron Physics Introduction...
Transcript of Lectures on Nuclear and Hadron Physics · Lectures on Nuclear and Hadron Physics Introduction...
Lectures on Nuclear and Hadron Physics
Lectures on Nuclear and Hadron Physics
J. A. Oller
Departamento de FısicaUniversidad de Murcia
July 27, 2011
Lectures on Nuclear and Hadron Physics
Contents
Outline Lecture III
1 Introduction
2 On the σ Resonance
3 On the ρ Resonance
4 Algebraic N/D
5 IAM
6 Mixing of Scalar Resonances
7 Large NC
8 Lightest Glueball
Lectures on Nuclear and Hadron Physics
Introduction
Introduction
1 The mesonic scalar sector has the vacuum quantum numbersJPC = 0++. Essential for the study of Spontaneous Breakingof Chiral Symmetry
2 The interactions with these quantum numbers are alsoessential for the masses of the lightest hadrons, thepseudo-Goldstone bosons (π, K , η), and ratio of light quarkmassesL. Roca, J.A.O. Eur. Phys. J. A 34 (2007) 371
3 In this sector meson interactions are really strongLarge unitarity loops (infrared enhancements and dynamicalgeneration of resonances)Channels that couple very strongly, e.g. ππ-KK , πη-KKResonances with large widths and in coupled channels: NOBreit-Wigner form, NO Vector-Meson-Dominance, NOnarrow-resonance approximation, etc
4 Okubo-Zweig-Izuka rule (large NC QCD) has large corrections:
Lectures on Nuclear and Hadron Physics
Introduction
NO ideal mixing multiplets
Simple quark model fails
Large unitarity loops
Lectures on Nuclear and Hadron Physics
σ Resonance
σ Resonance
The “lightest” hadronic resonance in QCD√sσ (MeV) Reference
469 − i 203 Oller, J.A.O. NPA620(1997)438;(E)NPA652(1999)307
441+16−8 − i 271+9
−12 Caprini,Colangelo,LeutwylerPRL96(2006)132001
484 ± 14 − i 255 ± 10 Garcıa-Martın, Pelaez, YndurainPRD76(2007)074034
456 ± 6 − i 241 ± 7 Albaladejo,J.A.O. PRL101(2008)252002
440+3−3 − i 258+2
−3 Z.-H.Guo, J.A.O., PRD in press
Lectures on Nuclear and Hadron Physics
σ Resonance
Connection of the σ with chiral symmetry
The lowest order chiral perturbation theory Lagrangian
L2 =F 2
π
4Tr
(∂µU∂µU† + U†χ+ χ†U
)
χ = 2B0(s + i p)
U = exp{
i√
2Φ/Fπ
}
Φ =
1√2π0 + 1√
6η8 π+ K+
π− − 1√2π0 + 1√
6η8 K 0
K− K0 − 2√
6η8
Lectures on Nuclear and Hadron Physics
σ Resonance
The lowest order JPC = 0++ ππ scattering amplitudeπ(p1)π(p2) → π(p3)π(p4)Mandelstam variables: s = (p1 + p2)
2 , t = (p1 − p3)2 and
u = (p1 − p4)2
T2 =s − m2
π/2
F 2π
It has a zero at s = m2π/2 (Adler zero)
For higher partial waves ℓ ≥ 1 the Adler zero is trivial. It is locatedat threshold.
Lectures on Nuclear and Hadron Physics
σ Resonance
Coming back to the N/D method without LHC:
T =
[∑
i
γi
s − si+ g(s)
]−1
To reproduce the Adler zero ONE CDD is needed
γ1 = F 2π
s1 = m2π/2
Lectures on Nuclear and Hadron Physics
σ Resonance
g(s) is the unitarity loop function
g(s) = g(s0) −∫ ∞
4m2π
ds ′ρ(s ′)
(s ′ − s0)(s ′ − s − iǫ)
=1
(4π)2
[a(µ) + log
m2π
µ2+ σ(s) log
σ(s) + 1
σ(s) − 1
]
Fitting ππ phase shifts
a(Mρ) = −0.79
T =
[1
T2+ g(s)
]−1
=T2
1 + T2g(s)= T2 − T2g(s)T2 + . . .
On the natural size for a(µ): First Unitarity Correction
T 22 g(s)
T2∼ a
16π2
s − m2π/2
F 2π
∼ as
16π2F 2π
|a| must be O(1). Dynamical Generated Resonance
Lectures on Nuclear and Hadron Physics
σ Resonance
Pole position of the σ
• Physical amplitudes have no pole positions in the complex splane with imaginary part because hermiticity would be violatedThe eigenvalues of a Hermitian operator like the Hamiltonian mustbe real.The pole positions for physical amplitudes should be real (boundstates)K. Gottfried, Quantum Mechanics, W.A. Benjamin
• Scattering amplitudes can be extended to unphysical Riemannsheets.There one can find poles corresponding to resonances or antiboundstates
Lectures on Nuclear and Hadron Physics
σ Resonance
Multi-valued function:
p =
√s
4− m2
π
Physical sheet: ℑ|p(s)| ≥ 0Second Riemann sheet: ℑ|p(s)| ≤ 0
+iǫ
−iǫ
I-SHEET
I-SHEET II-SHEET
II-SHEET
ℑg(s) = −|p|(s)8π
√s→ ℑgII (s) =
|p|(s)8π
√s
Lectures on Nuclear and Hadron Physics
σ Resonance
Schwartz’s reflection principle:
g(s + iǫ) − g(s − iǫ) = 2iℑg(s + iǫ) = −i|p|
4π√
s
g(s + iǫ) − gII (s + iǫ) = −i|p|
4π√
s
gII (s) = g(s) + i|p|
4π√
s
TII (s)=[T−1
2 + gII (s)]−1
TII (s)−1= T (s)−1 + i
|p|4π
√s
Lectures on Nuclear and Hadron Physics
σ Resonance
The σ pole position is located in TII (s)Because TII (s
∗) = TII (s)∗ there are indeed two poles at sσ and s∗σ
Laurent series
P
P′
sth R
Physical sheet, s variable
Because TII (s − iǫ) = T (s + iǫ) the pole P (with negativeimaginary part) is effectively much closer to the physical regionthan the pole P ′ (with positive imaginary part)
Lectures on Nuclear and Hadron Physics
σ Resonance
TII (s) =γ2
0
s − sP︸ ︷︷ ︸pole
+ γ1 + γ2(s − sP)︸ ︷︷ ︸non−resonant terms
+ . . .
The reason why the σ has been traditionally hard to establishexperimentally is due to the large contribution of non-resonantterms. Due to the need of cancellation for having the Adler zero.
sσ = (0.466 − i 0.224)2 GeV2
γ20 = 5.3 + i 7.7 GeV2
γ1 = −8.1 + i 36.9
γ2 = 1.1 + i 0.1 GeV2
Lectures on Nuclear and Hadron Physics
σ Resonance
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ρ
ρ
σ
σ
BW
δ00 BW
Lectures on Nuclear and Hadron Physics
σ Resonance
Breit Wigner formula
T ∝ e iδ sin δ =sin δ
cos δ − i sin δ=
1
cot δ − i
The partial wave T has a maximum at the resonance position, sothat sin δ = ±1 → cot δ = 0
cot δ = cot δ(ER) + (E − ER)(d cot δ/dE )|E=ER
(d cot δ/dE )|E=ER= −2/Γ
1
cot δ − i=
1
−2(E − ER)/Γ − i=
−Γ/2
E − ER + iΓ/2
Lectures on Nuclear and Hadron Physics
σ Resonance
If Γ ≪ ER then it is a good approximation
E + ER ≃ 2ER , s − sR ≃ (E − ER)2ER
−Γ/2
E − ER + iΓ/2≃ −ΓER
s − sR + iΓ ER
sR = E 2R
This is a Breit-Wigner (BW) formula for the amplitude
It is applied in many phenomenological studies to reproduce data
BW formula does not work well when typically the resonance isvery wide (like the σ) or it is located just on top of a threshold,even more when several channels couple
BW works very bad for the σ indeed.
Lectures on Nuclear and Hadron Physics
ρ resonance
The lightest preexisting resonance ρ(770)
Mρ = 770 MeV, Γρ = 150 MeV
N/D method without LHC:
T =
[∑
i
γi
s − si+ g(s)
]−1
To reproduce the (Adler) zero at threshold ONE CDD is needed
γ1 = 6F 2π
s1 = 4m2π
a = −6F 2π
M2ρ
16π2 ≃ −14.2
Lectures on Nuclear and Hadron Physics
ρ resonance
|a| ≫ 1 Preexisting Resonance (qq resonance)
Another argument to appreciate the difference:
a + logm2
π
µ2=
m2π
µ2→ µ = µ exp(−a/2)
with µ ∼ 800 MeV
µσ = 1.76 GeV
µρ = 970 GeV ∼ 1 TeV
To be compared with the natural scale ΛχPT = 4πFπ ∼ 1.2 GeV
The ρ pole dominates over the background in the Laurentexpansion
The BW formula for the ρ compares very well with the fullresult
Lectures on Nuclear and Hadron Physics
ρ resonance
Pole:
γ2R
s − sR
Its square root gives the couplingof the resonance to the ππ state|gσ→ππ| = 3.1 GeV|gρ→ππ| = 2.5 GeV
Width in terms of the coupling:
T = − g2
s − M2R + iMRΓR
Unitarity implies ℑT−1 = −|p|/8π√s at s = M2R
ΓR =g2|p|8πM2
R
Lectures on Nuclear and Hadron Physics
Algebraic N/D
Including other cuts in the algebraic N/D method
T =
[∑
i=1
γi
s − si+ g(s)
]−1
≡[R−1 + g(s)
]−1
A CDD at infinity corresponds to a constant
limsi→∞
γi
si= constant
The crossed cuts (LHC for equal mass scattering) are includedperturbatively in R
This is accomplished by matching order by order with T calculatedfrom the EFT, Tχ
U.-G. Meißner, J.A.O, Phys. Lett. B 500 (2001) 263
Lectures on Nuclear and Hadron Physics
Algebraic N/D
Tχ = T2 + T4 + . . . = [1 + Rg(s)]−1 R = R2 + R4 −R2gR2 + . . .
R2 = T2
R4 = T4 + R2gR2
and son on for higher orders
The resulting R has no RHC (that arises form g(s))
Lectures on Nuclear and Hadron Physics
Algebraic N/D
Production Process. E.g. Form factors
source
Final State InteractionsFSI
π
π π
π
� Production vertex• Full strong interaction vertex
Analogy with the scattering process
FSI
Lectures on Nuclear and Hadron Physics
Algebraic N/D
T = [1 + Rg ]−1 R → F = [1 + Rg ]−1 FUnitarity and analyticity can be used to obtain a more generalderivation of this resultOset, Palomar, J.A.O., Phys. Rev. D 63 (2001) 114009
1 R is known form the studies on the strong interactions2 F is fixed by matching with F calculated up to a given order
in the EFT
Pion and Kaon vector Form FactorsStrong amplitude: JPC = 1−−
Matching with EFT:= CHPT+Resonances at Leading OrderEcker, Gasser, Pich, de Rafael, Nucl. Phys. B 321 (1989) 311
R2 = T LOEFT
T LOEFT
∣∣∣11
=2|p|23F 2
π
[1 +
2G 2V
F 2π
s
M2ρ − s
]
Lectures on Nuclear and Hadron Physics
Algebraic N/D
Form Factors:Matching with NLO EFT result on Form Factors(CHPT+Resonances)
F1 = 1 +FV GV
F 2π
s
M2ρ − s
High energy QCD constraints were also imposed so that thesubtraction constants are determined (they are introduced fromthe strong interactions)
Lectures on Nuclear and Hadron Physics
Algebraic N/D
Lectures on Nuclear and Hadron Physics
Algebraic N/D
Pion and kaon Scalar Form Factors
U.-G. Meißner, J.A.O., Nucl. Phys. A 679 (2001) 671Strong amplitude: JPC = 0++
Matching with EFT:= CHPT at Leading OrderGasser, Leutwyler, Nucl. Phys. B 250 (1985) 465
R2 = T LOEFT
R11 =s − m2
π/2
F 2π
Form Factors
〈 0|ss|MM ′ 〉 , 〈 0|nn|MM ′ 〉
nn =1√2(uu + dd)
Lectures on Nuclear and Hadron Physics
Algebraic N/D
F nn1 =
√3
2
{1 +
4(L5 + 2L4)
F 2π
s +16(2L8 − L5)
F 2π
m2π
+8(2L6 − L4)
F 2π
(2m2K + 3m2
π) − m2π
32π2F 2π
− 1
3µn
}
Li : NLO CHPT Counterterms
Lectures on Nuclear and Hadron Physics
Algebraic N/D
J/Ψ → φ(w)ππ
Lectures on Nuclear and Hadron Physics
Algebraic N/D
J/Ψ → φ(w)ππ
T.A. Lahde, U.-G. Meißner, Phys. Rev.
D 74 (2006) 034021
L. Roca, J.E. Palomar, E. Oset, Nucl.
Phys. A 744 (2004) 127
Lectures on Nuclear and Hadron Physics
IAM
The Inverse Amplitude Method (IAM) with Coupled
Channels
Oset, Pelaez, J.A.O., Phys. Rev. Lett. 80 (1998) 3452• The IAM was introduced originally for the elastic case (only onechannel) inTruong, Phys. Rev. Lett. 61 (1998) 2526Dobado, Herrero, Truong Phys. Lett. B 235 (1990) 129Dobado, Pelaez, Phys. Rev. D 56 (1997) 3057• Its generalization to coupled channels was given inOset, Pelaez, J.A.O., Phys. Rev. Lett. 95 (2005) 172502Oset, Pelaez, J.A.O., Phys. Rev. D 59 (1999) 074001
From the CHPT expansion up to NLO of T we work out theexpansion of T−1
It reminds us of the effective range approximation of QuantumMechanics
Lectures on Nuclear and Hadron Physics
IAM
In this way if the amplitude has a pole its inverse has only azero
The number of free parameter is the same as in CHPT atNLO: L1, L2, L3, L4, L5, L7, 2L6 + L8 fitted to data for√
s ≤ 1.2 GeV
Explicit matrix language: Coupled Channels Tij corresponds toi → j
ℑT−1 = −ρ
ρi =|p|i
8π√
sθ(s − si )
T−1 = ℜT−1 − iρi
T−1 = [T2 + T4]−1 = T−1
2 ·[I + T4 · T−1
2
]−1
= T−12 ·
[I − T4 · T−1
2
]= T−1
2 · [T2 − T4] · T−12
Lectures on Nuclear and Hadron Physics
IAM
Figure: L = 0, I = 0 Figure: L = 0, I = 1
Lectures on Nuclear and Hadron Physics
IAM
Figure: L = 0, I = 1/2 Figure: L = 1, I = 1
Lectures on Nuclear and Hadron Physics
IAM
Figure: L = 1, I = 1/2 Figure: L = 0, I = 3/2
Lectures on Nuclear and Hadron Physics
IAM
Figure: L = 0, I = 2 Figure: L = 1, I = 0
Lectures on Nuclear and Hadron Physics
IAM
Resonances
Table III. Masses and partial widths in MeV
Channel
(I , J)Resonance
Mass
from pole
Width
from pole
Mass
effective
Width
effective
Partial
Widths
(0, 0) σ 442 454 ≈ 600 very large ππ − 100%
(0, 0) f0(980) 994 28 ≈ 980 ≈ 30ππ − 65%KK − 35%
(0, 1) φ(1020) 980 0 980 0(1/2, 0) κ 770 500 ≈ 850 very large Kπ − 100%(1/2, 1) K∗(890) 892 42 895 42 Kπ − 100%
(1, 0) a0(980) 1055 42 980 40πη − 50%KK − 50%
(1, 1) ρ(770) 759 141 771 147 ππ − 100%
Lectures on Nuclear and Hadron Physics
IAM
400600
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Figure: σ, f0(980)
Lectures on Nuclear and Hadron Physics
IAM
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σ
f0(980)f0(980)
Figure: σ, f0(980)
Lectures on Nuclear and Hadron Physics
IAM
700 720 740 760 780 800-100
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ρ(770)
Figure: ρ(770)
The gradient is oriented towards higher values of the mass thanthe pole position
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
Mixing angle of the lightest scalar nonet
SU(3) analysis
1961 Gell-Mann and Ne’eman introduced a classification ofhadrons based on SU(3) symmetry
SU(3) is the set of unitary transformations with determinant 1 inthree dimensions acting on the quark flavors or species
q(x) =
uds
For standard rotations one has multiplets like scalars, vectors,axials, etc
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
The QCD Lagrangian is invariant under SU(3) for equal quarkmasses
Different quark masses break SU(3) explicitly
q(x)
mu 0 00 md 00 0 ms
q(x)
ms ≫ mu,d
As a result SU(3) multiplets or irreducible representation mix
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
Well established octet+singlet of vector resonances. I = 0resonances mix.
replacements
ρ+
ρ0
ρ−φ, ω
K ∗+K ∗0
K ∗ 0 K ∗−
Ideal Mixing:
φ = ss , ω =1√2
(uu + dd
)
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
Well established octet+singlet of lightest pseudoscalars. I = 0states mix.
replacements
π+
π0
π−η, η′
K+ K 0
K 0 K−
η and η′ mixη = cos θP η8 − sin θP η1
η′ = sin θP η8 + cos θPs η1
θp ∼ −(15 − 20)o
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
What about the scalar resonances?
Mixing of the σ and f0(980)J.A.O. Nucl. Phys. A 727 (2003) 353Continuous movement from an SU(3) symmetric point with equalmasses m0: mi (λ) = mi + λ(m0 − mi ) with λ ∈ [0, 1]
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I=0
I=1
I=1/2
σ
f0(980)
a0(980)
κ
singlet
octet
ℑ√ s(M
eV)
ℜ√s (MeV)
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
a0(980), κ: Pure octet states
Clebsch-Gordan decomposition:
g(a0 → [KK ]I=1) = −√
3
10g8 , g(a0 → πη) =
1√5g8
g(κ→ Kπ) =3√20
g8 , g(κ→ Kη) =1√20
g8
Fraction CG Expg(a0→πη)
g(a0→[KK ]1)0.82 0.70 ± 0.04
g(a0→[KK ]1]g(a0→[Kπ]1/2)
0.82 1.10 ± 0.03
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
g(κ→[Kπ]1/2)
g(κ→Kη) 3 2.5 ± 0.3
|g8| = 8.7 ± 1.3 GeV
σ, f0(980) isosinglet states: Mixing
σ = cos θS1 + sin θS8
f0 = − sin θS1 + cos θS8
Ideal Mixing: cos θ2 = 2/3 = 0.67No consensus in the literature
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
g(σ → (ππ)0) = −√
3
4cos θg1 −
√3
10sin θg8 ,
g(σ → (KK )0) = −1
2cos θg1 +
1√10
sin θg8 ,
g(σ → (η8η8)0) =1
4cos θg1 −
1√10
sin θg8 ,
g(f0 → (ππ)0) =
√3
4sin θg1 −
√3
10cos θg8 ,
g(f0 → (KK )0) =1
2sin θg1 +
1√10
cos θg8 ,
g(f0 → (η8η8)0) = −1
4sin θg1 −
1√10
cos θg8 .
Lectures on Nuclear and Hadron Physics
Mixing of scalar resonances
g1
g8=
√8
5tan θ
θ = (+19 ± 5)o
g8 = (8.2 ± 0.8) GeV
g1 = (3.9 ± 0.8) GeV
The sign of θ can be fixed from the peak value at the f0(980)resonance of the scalar form factors
fn = 〈 0| 1√2(uu + dd)|KK 〉0
fs = 〈 0|ss|KK 〉0|fs/fn| ≫ 1
Lectures on Nuclear and Hadron Physics
Large NC
Large NC insight on the σ
QCD is a non-Abelian gauge QFT under SU(3)-color NC
In real life NC = 3, q(x)i , i = 1, . . . , 3
Dµ = ∂µ + ig√NC
Aµ
Fµν = ∂µAν − ∂νAµ + ig√NC
[Aµ,Aν ]
LQCD = −1
2TrFµνF
µν +
NF∑
k=1
ψk(i 6D −mk)ψk
A non-trivial NC → ∞ limit exists:G.‘t Hooft, Nucl. Phys. B 72 (1974) 461; B 75 (1974) 461E. Witten, Nucl. Phys. B 160 (1979) 57
Lectures on Nuclear and Hadron Physics
Large NC
The planar diagrams of QCD survive the large NC limit. All theothers are suppressed in large NC
β-function of QCD
µdg
dµ= −
(11
3− 2
3
NF
NC
)g3
16π2+ O(g5)
It has a well defined limit for NC → ∞Quark loops do not contribute NF/NC → 0 because insertion ofquark lines are suppressed in large NC
Lectures on Nuclear and Hadron Physics
Large NC
Fπ ∝√
Nc
〈0|Nc∑
i
qiγµγ5λ
aqi |πb(p)〉 = i δabFπ pµ
The operator has Nc termsA pion has the factor 1/
√Nc because of normalization∑Nc
i=1 qiqi/√
Nc
σ from LO Unitarized CHPT Oset,J.A.O. Phys. Rev. D 60 (1999)074023
F 2π
M2σ
+ gII (M2σ) = 0
M2σ ∝ F 2
π/gII (M2σ) ∼ Nc
Lectures on Nuclear and Hadron Physics
Large NC
According to the large NC rules for a qq meson:
M ∼ O(N0C ) , Γ ∼ 1/NC
which is not fulfilled by the σ (a0(980), κ) resonance.
Employing NLO CHPT
IAM results J.R.Pelaez PRL 92(2004)102001
Lectures on Nuclear and Hadron Physics
Large NC
IAM results J.R.Pelaez PRL 92(2004)102001
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σ does not correspond to suchbehavior
Lectures on Nuclear and Hadron Physics
Large NC
The QCD Lagrangian is invariant under U(3)L ⊗ U(3)R of quarkflavors
If UA(1) were spontaneously broken (like SU(3)A) then thereshould be an η0 with a mass <
√3mπ Weinberg PRD’75 but the η
is much heavier
The ninth axial singlet current has an anomalous divergence Adler
PR’69 Fujikawa PRD’80
Jµ (0)5 = qγµγ5q
∂µJµ (0)5 =
g2
16π2
1
NcTrc(GµνG
µν)
Large Nc QCD → nonet of pseudoscalars (Goldstone theorem)
Lectures on Nuclear and Hadron Physics
Large NC
The previous studies are based on SU(3) CHPT which does nothave the right number of degrees of freedom in large NC
It has only 8 Goldstone bosons (π, K , η8) but not 9 (including theη1) as large NC U(3) symmetry requires
Evolution of pseudoscalar masses with NC Z.-H. Guo, J.A.O.,PRD(2011) in press
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Lectures on Nuclear and Hadron Physics
Large NC
We make use U(3) CHPT
Combined power expansion in light quark masses, soft externalmomenta and 1/NC
Di Vecchia, Veneziano, NPB’80Rosenzweig, Schechter, Trahern, PRD’80Witten, Ann.Phys.’80 Herrera-Siklody, Latorre, Pascual, Taron,NPB’97
δ-counting: δ ∼ O(p2) ∼ mq ∼ 1/NC
Lectures on Nuclear and Hadron Physics
Large NC
σ-Pole Trajectory
200
300
400
500
600
700
800
900
1000
1100
1200
100 200 300 400 500 600 700
mimic SU(3)vector reduced
full result
Γ/2
(MeV
)
Mass (MeV)
NC = 3
σ
Oller, Oset PRD’99 M2S ∝ f 2 ∝ Nc
Pelaez et al ’04,’06,. . . ,’10 σ Mass always increases with Nc
At the two-loop order it moves to a pole with zero width at 1 GeV.
We also obtain such a pole but it comes from the bare scalarsinglet MS1 ≃ 1 GeV (At Nc = 3 it contributes to the f0(980).)
Lectures on Nuclear and Hadron Physics
Large NC
1
M2V − t
→ 1
M2V
,
NLO local terms in χPT. Vector Reduced case (Blue Triangles)
The σ-pole trajectory is then more similar to that of the IAMone-loop study Pelaez PRL’04
Sensitivity to higher order local terms.
When full vector propagators are kept their crossed exchangecontributions cancel mutually with those from crossed loops alongthe RHC (
√s . 1 GeV) Oller, Oset PRD’99
For increasing Nc loops are further suppressed and this cancellationis spoilt: More sensitivity to the LHC contributions.
The σ pole blows-up in the complex plane (not qq). Dynamicallygenerated resonance.
Lectures on Nuclear and Hadron Physics
Large NC
There is no influence on the σ NC pole trajectory by the largereduction of the η mass because the σ does couple very little tostates with η and η′
For the other scalar resonances this is not the case because theycouple strongly with such states
We also obtain the standard qq behavior for the lightest octet ofvector resonances ρ, K ∗(892) and φ(1020)
The f0(980), f0(1370), a0(1450), K ∗0 (1430) evolve at NC → ∞ to
preexisting states.
Lectures on Nuclear and Hadron Physics
Large NC
Mimic SU(3) case: Mixing is set to zero and η1 is kept in the loops. η8,
η1 masses are frozen. Differences highlight the role of η′. Differences for
the σ case are negligible.
0
10
20
30
40
50
960 980 1000 1020 1040 1060 0
40
80
120
160
200
1320 1340 1360 1380 1400
mimic SU(3)vector reduced
full result
0
20
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100
120
140
1320 1340 1360 1380 1400 1420 1440 1460 0
20
40
60
80
100
120
140
1340 1360 1380 1400 1420 1440
4567
Γ/2
(MeV
)
Γ/2
(MeV
)
Γ/2
(MeV
)
Γ/2
(MeV
)
Mass (MeV)Mass (MeV)
Mass (MeV)Mass (MeV)
NC = 3
NC = 3
NC = 3NC = 3
NC = 3
NC = 3
NC = 3NC = 3
NC = 3f0(980)
f0(1370)
K∗
0(1430)
a0(1450)
Lectures on Nuclear and Hadron Physics
Large NC
ρ(770), K ∗(892)
0
10
20
30
40
50
60
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760 770 780 790 800
mimic SU(3)full result
0
5
10
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30
890 895 900 905 910
mimic SU(3)full result
Γ/2
(MeV
)
Γ/2
(MeV
)
Mass (MeV)Mass (MeV)
NC = 3NC = 3
ρ(770) K∗(892)
qq trajectories: Mass O(N0c ) and Width O(1/Nc) Pelaez PRL’04
Above Nc = 13 the Kη threshold becomes lighter than theK ∗(892) mass (kink in the residue to Kη at Nc = 14).For the φ(1020) the situation is the same. There is sensitivity tothe slight movement of the nearby KK threshold with Nc .
Lectures on Nuclear and Hadron Physics
Lightest Glueball
On the lightest Glueball
M.Albaladejo, J.A.O., Phys. Rev. Lett. 101 (2008) 252002
A new state of matter made only of gluonsDue to the non-Abelian nature of QCD gluons interact betweeneach other and can make hadrons without valence quarks
Predicted by quenched lattice QCD with M = (1.66 ± 0.05) GeV
Experimentally one finds the f0(1500) and f0(1710) near thispredicted mass.
Multi-coupled-channel study:
I = 0: ππ, KK , ηη, σσ, ηη′, ρρ, ωω, ηη′, ωφ, φφ, K ∗K∗,
a1(1260)π, π⋆π
I = 1/2, I = 3/2: Kπ, Kη and Kη′ Jamin, Pich, J.A.O.,NPB 587 (2000) 331
Lectures on Nuclear and Hadron Physics
Lightest Glueball
σσ channel amplitudes. Pion rescattering
We want to obtain σσ amplitudes starting from chiral Lagrangians.σ is S-wave ππ interaction, |σ〉 = |ππ〉0, Oset,J.A.O., NPA620’97
Pion rescattering is given by the factorD−1(s) = (1 + t2G (s))−1, with:
t2 =s−m2
π/2
f 2π
basic ππ → ππ amplitude.
(4π)2G (s) = α+ logm2
π
µ2 − σ(s) log σ(s)−1σ(s)+1 , two pion loop.
Isolate the transition amplitude Ni→σσ:
limsi→sσ
Ti→(ππ)0(ππ)0
DII (s1)DII (s2)=
Ni→σσg2σππ
(s1 − sσ)(s2 − sσ)
σ
σ
i
Lectures on Nuclear and Hadron Physics
Lightest Glueball
Around the σ pole DII (s)−1 = (1 + t2G (s))−1 ≈ α0
s−sσ+ · · · ,
then:
Na→(σσ)0= Ta→(ππ)0(ππ)0
(α0
gσππ
)2
To calculate (α0/gσππ)2, consider ππ elastic scattering,
T =t2(s)
1 + t2(s)G (s)≈ − g2
σππ
s − sσ+ · · ·
So we can write, using that at the σ polegII (sσ) = −1/t2(sσ):
(α0
gσππ
)2
=F 2
π
1 − G′
II (sσ)F 2πt2(sσ)2
≈ 1.1F 2π
In conclusion, we follow a novel method to calculateamplitudes involving σσ, through:
Na→(σσ)0= Ta→(ππ)0(ππ)0
(α0
gσππ
)2
Lectures on Nuclear and Hadron Physics
Lightest Glueball
√s (MeV)
η0 0
15001000500
1.5
1
0.5
0
425350275
30
20
10
0
δ0 0
(deg
)
15001000500
450
360
270
180
90
0
ππ → ππ: Although reachingenergies of 2 GeV, description oflow energy data is still quite good
Lectures on Nuclear and Hadron Physics
Lightest Glueball
√s (MeV)
˛ ˛
S1,2
˛ ˛
15001000
1
0.5
0
Cohen et al.
Etkin et al.
δ1,2
(deg
)
15001000360
270
180ππ → KK : Near threshold,Cohen data are favored.
Lectures on Nuclear and Hadron Physics
Lightest Glueball
√s (MeV)
˛ ˛
S1,5
˛ ˛
2
1800170016001500
0.1
0
√s (MeV)
˛ ˛
S1,3
˛ ˛
2
200015001000
0.2
0.1
0
ππ → ηη,ηη′: In good agreementfor a low weight on χ2. Inaddition, the data areunnormalized.
Lectures on Nuclear and Hadron Physics
Lightest Glueball
√s (MeV)
A
1750150012501000750
250
200
150
100
50
√s (MeV)
φ(d
eg)
1750150012501000750
180
150
120
90
60
30
0 I = 1/2 K−π+ → K−π+
amplitude and phase from LASS.
Lectures on Nuclear and Hadron Physics
Lightest Glueball
No parameterization
We determine interaction kernels from Chiral Lagrangians, avoidingad hoc parameterizations
Less free parameters
We have less free parameters, because of our chiral approach andour treatment of σσ amplitude
Higher energies
We have included enough channels to get at 2 GeV.
Compare with: Lindenbaum,Longacre PLB274’92; Kloet, Loiseau,
ZPA353’95; Bugg, NPB471’96
Lectures on Nuclear and Hadron Physics
Lightest Glueball
Spectroscopy. Pole content: Summary
All the scalar resonances with mass up to 2 GeV are reproduced
I = 0 Poles (MeV) Effect M (PDG) Γ (PDG/BESII)
σ ≡ f0(600) 456 ± 6 − i 241 ± 7
f0(980) 983 ± 4 − i 25 ± 4 980 ± 10 40 − 100
f0(1370) 1466 ± 15 − i 158 ± 12 1200 − 1500 200 − 500
f0(1500) 1602 ± 15 − i 44 ± 15 1505 ± 6 109 ± 7
f0(1710) 1690 ± 20 − i 110 ± 20 1724 ± 7 137 ± 8
f0(1790) 1810 ± 15 − i 190 ± 20 1790+40−30 270+30
−60
I = 1/2 Poles (MeV) Effect M (PDG) Γ (PDG)
708 ± 6 − i 313 ± 10 κ ≡ K∗0 (800) − −
1435 ± 6 − i 142 ± 8 K∗0 (1430) 1414 ± 6 290 ± 21
1750 ± 20 − i 150 ± 20 K∗0 (1950) − −
Lectures on Nuclear and Hadron Physics
Lightest Glueball
Spectroscopy. Couplings
f 0(1370) (I = 0) K ∗0 (1430) (I = 1/2)
Coupling bare final Coupling bare final
gπ+π− 3.9 3.59 ± 0.18 gKπ 5.0 4.8
gK0K0 2.3 2.23 ± 0.18 gKη 0.7 0.9
gηη 1.4 1.70 ± 0.30 gKη′ 3.4 3.8
gηη′ 3.7 4.00 ± 0.30
gη′η′ 3.8 3.70 ± 0.40
Bare coupling are those of S(1)8 , with M
(1)8 = 1.3 GeV.
The first preexisting scalar octet is a pure one, not mixed withthe nearby f0(1500) and f0(1710)
Lectures on Nuclear and Hadron Physics
Lightest Glueball
Spectroscopy. Couplings
Coupling (GeV) f 0(1500) f0(1710)
gπ+π− 1.31 ± 0.22 1.24 ± 0.16
gK0K0 2.06 ± 0.17 2.00 ± 0.30
gηη 3.78 ± 0.26 3.30 ± 0.80
gηη′ 4.99 ± 0.24 5.10 ± 0.80
gη′η′ 8.30 ± 0.60 11.7 ± 1.60
Coupling (GeV) f 0(1500) f0(1710)
gss 11.5 ± 0.5 13.0 ± 1.0
gns −0.2 2.1
gnn −1.4 1.2
gss/6 1.9 ± 0.1 2.1 ± 0.2
This pattern suggests a suppression in uu and dd production.With a pseudoscalar mixing angle sinβ = −1/3 for η and η′:
Lectures on Nuclear and Hadron Physics
Lightest Glueball
gη′η′ =2
3gss +
1
3gnn +
2√
2
3gns ,
gηη′ = −√
2
3gss +
√2
3gnn +
1
3gns ,
gηη =1
3gss +
2
3gnn −
2√
2
3gns .
If chiral suppression M.S.Chanowitz, PRL95(’05)172001;
98(’07)149104 operates:The coupling of the glueball to qq is proportional to mq
|gss | ≫ |gnn||gns | ≫ |gnn| expected from OZI rule
Lectures on Nuclear and Hadron Physics
Lightest Glueball
Consider KK coupling
Valence quarks: K 0 corresponds to∑3
i=1 siuj/√
3, andanalogously K 0
Production of color singlet ss requires combination of colorwave functions of K 0, K 0
Decompose si sj = δji ss/3 + (si s
j − δji ss/3) and similarly uiuj
Only ssuu contributes (factor 1/3) and ssss has an extrafactor two compared to ssuu, so one expects gK0K0 = gss/6
Lectures on Nuclear and Hadron Physics
Lightest Glueball
The f0(1710) resonance corresponds to the lightest scalar glueball.We find a really good agreement with the mechanism of chiralsuppression from QCD proposed by Chanowitz with our exhaustivemultichannel hadronic calculation.
It is a pure glueball not mixed with any other resonance
f0(1370) is almost purely octet states
f0(1500) is made up of the sum of the poles that give rise to thef0(1370) and f0(1710) together with a strong cusp effect from theηη′ threshold.